Molecular Orbital Theory and Bonding/Antibonding States

The Born–Oppenheimer approximation hands molecular physics its central simplification: freeze the nuclei, and the electrons see a fixed two-centre potential whose eigenvalues are the electronic energy. But two centres already make the problem unsolvable in closed form — everything past the one-electron ion $\mathrm{H_2^+}$ is analytically intractable. Molecular orbital (MO) theory is the working answer: it builds each electron's wavefunction as a linear combination of the atomic orbitals it already knows — LCAO — and lets the variational principle pick the mixing that minimises the energy. The stationary combinations always come in pairs straddling the isolated atomic level: one pulled down, one pushed up. That splitting is the whole story. The in-phase combination is a bonding orbital: constructive interference piles electron probability between the nuclei, and that shared negative charge screens the nuclear repulsion and glues the molecule together. The out-of-phase combination is an antibonding orbital, marked with an asterisk: destructive interference carves a node between the nuclei and raises the energy. Fill the orbitals from the bottom by aufbau, and two headline numbers fall out — the bond order $\tfrac{1}{2}(N_\mathrm b-N_\mathrm a)$ , which tracks bond strength, length, and stiffness, and the count of unpaired spins, which fixes the magnetism. MO theory is also the electronic backbone of spectroscopy: the $\sigma/\pi$ , gerade/ungerade, and $\pm$ labels it assigns become the term symbols, and the rules built from them are exactly the selection rules a photon must obey.

Key takeaways

Molecular orbitals are linear combinations of atomic orbitals (LCAO): the in-phase sum $\psi_A+\psi_B$ is a bonding MO with charge piled between the nuclei; the difference $\psi_A-\psi_B$ is an antibonding MO with a node between them.
Variation over a symmetric two-orbital basis gives $E_\pm=(\alpha\pm\beta)/(1\pm S)$ : the resonance integral $\beta<0$ makes the split, and positive overlap $S$ makes the antibonding level rise more than the bonding level falls.
Filling MOs by increasing energy fixes the configuration; the bond order $\tfrac12(N_\mathrm b-N_\mathrm a)$ predicts whether a molecule binds and how strong, short, and stiff the bond is.
Hund's rule inside degenerate $\pi$ orbitals fixes the magnetism — $\mathrm{O_2}$ has two unpaired $\pi_g^*$ electrons and a paramagnetic $^3\Sigma_g^-$ ground state that Lewis structures get wrong.
The $\sigma/\pi/\delta$ , g/u, and $\pm$ labels of the occupied MOs are the origin of molecular term symbols and of the electric-dipole selection rules for electronic transitions.
Key idea: two atomic orbitals interfering in phase give a bonding MO (energy lowered, density between the nuclei); out of phase give an antibonding MO (energy raised, a node between the nuclei) — and filling these by aufbau yields bond order and magnetism.

The core insight

Two atomic orbitals interfering in phase give a bonding MO (energy lowered, density between the nuclei); out of phase give an antibonding MO (energy raised, a node between the nuclei). Filling these by aufbau yields bond order and magnetism.

Where this lives today

The bonding/antibonding language is the everyday grammar of molecular spectroscopy and photochemistry. Absorbing an ultraviolet or visible photon promotes an electron from a filled MO into an empty, usually antibonding one — a $\pi\to\pi^*$ or $n\to\pi^*$ transition — and whether the excited state is bound or dissociative is read straight off the orbitals' bonding character. Photoelectron spectroscopy goes further, reading MO binding energies directly, and the ordering it reveals — including the $\sigma_{2p}$ / $\pi_{2p}$ crossover between $\mathrm{N_2}$ and $\mathrm{O_2}$ — is a routine confirmation of the diagram. The same orbitals drive technology. The frontier MOs, HOMO and LUMO, set the band gaps of organic LEDs and photovoltaics, so molecular design for displays and solar cells is the engineering of antibonding-orbital energies; and quantum-chemistry codes from Hartree–Fock to density-functional theory are industrial-scale descendants of the LCAO secular problem in Derivation 1, now solved over thousands of basis functions instead of two.

The physical picture

Treat two identical atomic orbitals brought together as two pendulums coupled by a weak spring: coupling splits their shared frequency into a lower symmetric in-phase mode and a higher antisymmetric out-of-phase one. Bonding and antibonding molecular orbitals are exactly this two-state splitting, its size set by the resonance integral $\beta$ . The glue is interference. The bonding orbital $\psi_+\propto\psi_A+\psi_B$ has a density carrying a cross term $+2\psi_A\psi_B$ that is positive precisely between the nuclei; an electron there feels both attractions at once and spreads its wavefunction, lowering the kinetic energy too. The antibonding orbital $\psi_-\propto\psi_A-\psi_B$ flips that sign: density is gouged out, a node appears at the midpoint of a homonuclear bond, and the electron is banished to the flanks. The first figure plots both densities along the bond axis for two hydrogen 1s orbitals.

Figure

Constructive vs destructive interference along the bond axis: two hydrogen 1s orbitals at separation R = 2a₀ ≈ 1.06 Å with overlap S ≈ 0.59 — the in-phase sum ψ₊ piles charge between the nuclei while the out-of-phase ψ₋ has a node exactly at the midpoint

Magnetism falls straight out of filling. When the highest occupied level is a doubly degenerate $\pi$ pair, Hund's rule places one electron in each component with parallel spins, leaving a net spin. This is why $\mathrm{O_2}$ — two electrons in degenerate $\pi_g^*$ orbitals — is a triplet drawn to a magnet, a fact the shared-pair Lewis double bond gets flatly wrong. Bond order, meanwhile, is a one-integer shorthand for the whole potential curve: higher order means a deeper well and a shorter, stiffer, more strongly bound diatomic. The second figure is the payoff — the filled $\mathrm{O_2}$ diagram, read off in a glance.

Figure

The O₂ molecular-orbital diagram filled with 12 valence electrons: the last two occupy the degenerate antibonding π* pair with parallel spins by Hund's rule, giving bond order 2, two unpaired electrons, and a paramagnetic ³Σg⁻ ground state that Lewis structures cannot reach

The core equations

The starting move is the LCAO ansatz: a molecular orbital is a weighted sum of atomic orbitals on each nucleus. For a homonuclear diatomic, symmetry forces equal weights, so only the relative sign is free — the two normalised combinations below are the bonding sum and the antibonding difference.

\psi_\pm=\frac{1}{\sqrt{2(1\pm S)}}\big(\psi_A\pm\psi_B\big) \qquad (1)

where: $\psi_A,\psi_B$ — atomic orbitals on nuclei A and B; $\psi_+$ — bonding MO (in-phase, $\sigma_g$ -type); $\psi_-$ — antibonding MO (out-of-phase, $\sigma_u^*$ -type); $S$ — overlap integral, $0<S<1$ . The normalisation prefactor already carries the bonding/antibonding asymmetry. Three integrals set every number that follows: the overlap $S$ measures how much the atomic orbitals share space, the Coulomb integral $\alpha$ is the energy of an electron in one atomic orbital feeling the whole molecular field, and the resonance integral $\beta$ is the coupling — an electron's amplitude to hop between centres — that produces the split.

S=\int\psi_A^*\psi_B\,\mathrm dV,\qquad \alpha=\int\psi_A^*\hat H\psi_A\,\mathrm dV,\qquad \beta=\int\psi_A^*\hat H\psi_B\,\mathrm dV \qquad (2)

where: $\hat H$ — one-electron molecular Hamiltonian at fixed $R$ ; $\alpha$ — Coulomb integral, the perturbed atomic level (negative); $\beta$ — resonance integral, the interference coupling (negative for orbitals overlapping in phase). With $\beta=0$ there is no bond; $\beta$ is where chemistry enters. Feeding the ansatz through the variational principle and solving the $2\times2$ secular problem gives both stationary energies in closed form — the workhorse result, obtained in Derivation 1.

E_\pm=\frac{\alpha\pm\beta}{1\pm S} \qquad (3)

where: $E_+$ — bonding energy, below $\alpha$ since $\beta<0$ ; $E_-$ — antibonding energy, above $\alpha$ . The overlap $S$ in the denominators breaks the symmetry: the antibonding level rises by more than the bonding level falls. With the one-electron levels ordered by energy, a many-electron molecule is built by aufbau and summarised by one count — half the excess of bonding over antibonding electrons.

\text{bond order}=\tfrac12\big(N_\mathrm b-N_\mathrm a\big) \qquad (4)

where: $N_\mathrm b$ — electrons in bonding MOs; $N_\mathrm a$ — electrons in antibonding MOs. Values $1,2,3$ are single, double, triple bonds; $0$ means no binding; fractional values (like $2.5$ for $\mathrm{O_2^+}$ ) are physical. For second-row diatomics the valence order itself carries physics: it changes across the row because of $s$ – $p$ mixing. For $\mathrm{Li_2}$ through $\mathrm{N_2}$ the mixing is strong and pushes $\sigma_{2p}$ above the $\pi_{2p}$ pair; for $\mathrm{O_2}$ , $\mathrm{F_2}$ , $\mathrm{Ne_2}$ the mixing is weak and the natural order $\sigma_{2p}<\pi_{2p}$ is restored.

\underbrace{\sigma_{2s}<\sigma_{2s}^*<\pi_{2p}<\sigma_{2p}<\pi_{2p}^*<\sigma_{2p}^*}_{\mathrm{Li_2}\text{–}\mathrm{N_2}\ (\text{strong }s\text{–}p)},\qquad \underbrace{\sigma_{2s}<\sigma_{2s}^*<\sigma_{2p}<\pi_{2p}<\pi_{2p}^*<\sigma_{2p}^*}_{\mathrm{O_2}\text{–}\mathrm{Ne_2}\ (\text{weak})} \qquad (5)

where: $\sigma,\pi$ — MOs with angular-momentum projection $\Lambda=0,1$ about the axis; $*$ — antibonding. The crossover is immaterial to bond order and to $\mathrm{O_2}$ 's magnetism, since those orbitals are filled either way. Interference shows up most directly in the probability density. Squaring the normalised MOs isolates the cross term that is the physical origin of bonding.

|\psi_\pm|^2=\frac{|\psi_A|^2+|\psi_B|^2\pm2\,\psi_A\psi_B}{2(1\pm S)} \qquad (6)

where: $2\psi_A\psi_B$ — the interference density; $+$ piles it between the nuclei (bonding), $-$ removes it and forces a node (antibonding) — not any change in the Coulomb law. Homonuclear molecules have an inversion centre, so every MO carries a definite parity — a symmetry quantum number that survives into the term symbol and the selection rules.

\hat P\,\psi_g=+\psi_g,\qquad \hat P\,\psi_u=-\psi_u \qquad (7)

where: $\hat P$ — inversion through the molecular centre; $g$ (gerade, even) — e.g. $\sigma_g$ or $\pi_g^*$ ; $u$ (ungerade, odd) — e.g. $\sigma_u^*$ or $\pi_u$ . Parity is defined only for centrosymmetric molecules and generates the Laporte rule $g\leftrightarrow u$ for electric-dipole transitions.

Derivation AThe bonding and antibonding energies from the secular equation

The goal is to obtain the closed-form energies $E_\pm=(\alpha\pm\beta)/(1\pm S)$ that underlie every MO diagram, from nothing but the two-orbital trial function and the variational principle. Minimising the energy over the mixing coefficients turns stationarity into a pair of linear secular equations whose non-trivial solution demands a vanishing determinant, which the homonuclear symmetry collapses to two clean roots — showing where $\beta$ makes the split and $S$ makes it asymmetric.

Step 1. Take $\psi=c_A\psi_A+c_B\psi_B$ , form the Rayleigh quotient $\langle E\rangle=\langle\psi|\hat H|\psi\rangle/\langle\psi|\psi\rangle$ , and impose $\partial\langle E\rangle/\partial c_A=\partial\langle E\rangle/\partial c_B=0$ . Why: the variational theorem makes the best coefficients the ones that render the energy stationary, converting "guess the wavefunction" into linear algebra.

Step 2. With $H_{AA}=H_{BB}=\alpha$ , $H_{AB}=\beta$ , $S_{AB}=S$ , stationarity becomes $(\alpha-E)c_A+(\beta-ES)c_B=0$ and $(\beta-ES)c_A+(\alpha-E)c_B=0$ . Why: each derivative gives one homogeneous equation, with $E$ appearing wherever $S$ multiplies the energy — this is what makes it a generalised eigenvalue problem.

Step 3. A non-trivial solution needs the determinant to vanish, $\begin{vmatrix}\alpha-E&\beta-ES\\\beta-ES&\alpha-E\end{vmatrix}=0$ , i.e. $(\alpha-E)^2=(\beta-ES)^2$ . Why: homogeneous equations with a non-zero solution require a singular matrix, and the symmetric $2\times2$ form factorises by taking square roots.

Step 4. Then $\alpha-E=\pm(\beta-ES)$ gives $E_\pm=(\alpha\pm\beta)/(1\pm S)$ , with $c_A=\pm c_B$ recovering the bonding and antibonding combinations of (1). Why: the $+$ root ties to symmetric coefficients (bonding), the $-$ root to antisymmetric ones (antibonding), so the algebra reproduces the physical pairing exactly. *As a check: as $S\to0$ this reduces to the symmetric splitting $E_\pm=\alpha\pm\beta$ , confirming that overlap is the sole source of the asymmetry.*

Go deeper · Derivation B · 1 minWhy the antibonding level rises more than the bonding level falls

Derivation BWhy the antibonding level rises more than the bonding level falls

The goal is to prove the asymmetry that plain energy diagrams hide: with real, positive overlap, filling both members of a bonding/antibonding pair is not neutral but net repulsive. We measure each level relative to the atomic energy $\alpha$ and compare the two shifts term by term; the culprit is the $1\pm S$ denominator, and the payoff is $\mathrm{He_2}$ — equal bonding and antibonding occupancy, yet unbound.

Step 1. The bonding shift is $\Delta E_+=E_+-\alpha=\dfrac{\beta-\alpha S}{1+S}$ , negative for the usual $|\beta|>|\alpha|S$ : the level drops below $\alpha$ . Why: subtracting the reference $\alpha$ isolates the pure stabilisation, over the larger denominator $(1+S)$ .

Step 2. The antibonding shift is $\Delta E_-=E_--\alpha=\dfrac{-\beta+\alpha S}{1-S}$ , positive because $-\beta>0$ dominates: the level rises above $\alpha$ . Why: the same-magnitude numerator now sits over the smaller denominator $(1-S)$ .

Step 3. Comparing, $|\Delta E_-|-|\Delta E_+|=(-\beta+\alpha S)\!\left(\dfrac{1}{1-S}-\dfrac{1}{1+S}\right)=(-\beta+\alpha S)\dfrac{2S}{1-S^2}>0$ . Why: both factors are positive for $0<S<1$ , so the smaller $(1-S)$ denominator makes the antibonding rise exceed the bonding fall — destabilisation always wins.

Step 4. For $\mathrm{He_2}$ , the configuration $(\sigma_{1s})^2(\sigma_{1s}^*)^2$ gives bond order $\tfrac12(2-2)=0$ , and the filled antibonding pair outweighs the bonding pair, so the net effect is repulsive. Why: bond order counts electrons, but the asymmetry supplies the energetics — which is why $\mathrm{He_2}$ exists only as an extraordinarily weak van der Waals dimer. *As a check: the bracket equals $2S/(1-S^2)>0$ on $0<S<1$ , confirming the inequality holds for any real overlap; and the $\mathrm{He_2}$ conclusion matches experiment — the only bound state of $\mathrm{He_2}$ is a van der Waals dimer only a few mK deep.*

Limiting cases

Case	Result	Meaning
Large separation $R\to\infty$	$S\to0,\ \beta\to0,\ E_\pm\to\alpha$	Overlap and coupling vanish; both MOs collapse to the atomic level — no bond.
Zero-overlap idealisation $S=0$	$E_\pm=\alpha\pm\beta$	Symmetric splitting; simple, but misses the destabilisation asymmetry.
Finite overlap $S>0$ (real molecule)	$E_+=\dfrac{\alpha+\beta}{1+S},\ E_-=\dfrac{\alpha-\beta}{1-S}$	Antibonding destabilised more than bonding stabilised; filling both ( $\mathrm{He_2}$ ) is net repulsive.
Heteronuclear, $\|\alpha_A-\alpha_B\|\gg\|\beta\|$	MOs localise on the lower-energy atom	Bond becomes ionic/polar; interference stabilisation shrinks; g/u labels no longer apply.
Degenerate frontier level, half-filled	Hund's rule → maximum spin	Parallel spins give a net moment — the origin of $\mathrm{O_2}$ 's paramagnetism.

Common trapBonding versus antibonding is about interference, not a change in the Coulomb force — and filling a bonding/antibonding pair is net destabilising, not neutral

Two traps travel together. First, an antibonding orbital does not arise because the nuclei suddenly repel; the Coulomb law is unchanged. What changes is the interference of the electron amplitude — constructive piles density between the nuclei (bonding), destructive carves a node there (antibonding). Second, a filled bonding and antibonding pair does not cancel to zero cost: because the overlap $S>0$ sits in the denominators of $E_\pm=(\alpha\pm\beta)/(1\pm S)$ , the antibonding level is pushed up more than the bonding level is pulled down, so the pair is net repulsive. That single fact is why $\mathrm{He_2}$ — bond order zero, two bonding and two antibonding electrons — is unbound rather than merely weak.

Worked examples

1 · Oxygen is a paramagnetic double bond (the confidence builder)

Fill the $\mathrm{O_2}$ diagram, find its bond order and unpaired-electron count, and state the ground term. Oxygen has $Z=8$ , so $\mathrm{O_2}$ has 16 electrons; use the $\mathrm{O_2}$ order $\sigma_{2s}<\sigma_{2s}^*<\sigma_{2p}<\pi_{2p}<\pi_{2p}^*$ .

Step 1. The four core electrons fill $(\sigma_{1s})^2(\sigma_{1s}^*)^2$ ; this bonding/antibonding pair cancels and is dropped, leaving 12 valence electrons.

Step 2. The $2s$ MOs fill as $(\sigma_{2s})^2(\sigma_{2s}^*)^2$ — again a cancelling pair; 8 valence electrons remain.

Step 3. The bonding $2p$ MOs fill in order, $(\sigma_{2p})^2(\pi_{2p})^4$ , using six electrons to fill one $\sigma$ and both degenerate $\pi$ bonding orbitals; 2 remain.

Step 4. The last two enter the degenerate antibonding pair; by Hund's rule they occupy the components singly with parallel spins, $(\pi_{2p_x}^*)^1(\pi_{2p_y}^*)^1$ , giving $S=1$ and two unpaired electrons. With $N_\mathrm b=2+4+2=8$ and $N_\mathrm a=2+2=4$ , the bond order is $\tfrac12(8-4)=2$ — a double bond — and the ground state is the triplet $^3\Sigma_g^-$ , so $\mathrm{O_2}$ is paramagnetic, as seen when liquid oxygen bridges a magnet's poles. Bond order 2 also places its dissociation energy ( $\approx498$ kJ/mol) between $\mathrm{N_2}$ ( $\approx945$ ) and $\mathrm{F_2}$ ( $\approx158$ ).

2 · How far the levels split — a numeric two-level estimate

Take representative one-electron integrals for a homonuclear $\sigma$ bond: $\alpha=-13.6\ \mathrm{eV}$ , $\beta=-5.0\ \mathrm{eV}$ , $S=0.20$ . Find the two energies, the splitting, and confirm the asymmetry.

Step 1. Bonding: $E_+=\dfrac{\alpha+\beta}{1+S}=\dfrac{-18.6}{1.20}=\mathbf{-15.5\ eV}$ , a stabilisation of $\Delta E_+=-15.5-(-13.6)=\mathbf{-1.9\ eV}$ .

Step 2. Antibonding: $E_-=\dfrac{\alpha-\beta}{1-S}=\dfrac{-8.6}{0.80}=\mathbf{-10.75\ eV}$ , a destabilisation of $\Delta E_-=-10.75-(-13.6)=\mathbf{+2.85\ eV}$ .

Step 3. The total gap is $E_--E_+=\mathbf{4.75\ eV}$ , and the asymmetry is explicit: the antibonding level rises by $2.85$ eV while the bonding level falls by only $1.9$ eV. If both were doubly occupied the net change would be $2(-1.9)+2(+2.85)=+1.9$ eV — repulsive, the $\mathrm{He_2}$ story in numbers. The same counting ranks the series: $\mathrm{N_2}$ (order 3, $1.10$ Å), $\mathrm{O_2}$ (order 2), $\mathrm{F_2}$ (order 1, $1.42$ Å), and $\mathrm{O_2^+}$ (order $2.5$ ) — where removing an antibonding $\pi_{2p}^*$ electron shortens the bond from $1.21$ Å to $1.12$ Å.

Physics of the portrait

Molecular orbital theory has an unusually clean origin: within a few years around 1930 a handful of physicists asked what quantum mechanics says when two atoms share their electrons, and built essentially the theory taught today.

Origins · 1 minRobert Mulliken and how this was discovered

Robert Mulliken

1896–1986 · molecular orbital theory and molecular electronic structure

Robert Sanderson Mulliken turned spectroscopic pattern-recognition into a theory of the molecule and gave the field its vocabulary — he coined the term "molecular orbital" in 1932. With Friedrich Hund he developed the picture in which electrons occupy orbitals belonging to the whole molecule rather than to individual bonds, a direct challenge to the localised shared-pair valence-bond theory of Heitler, London, and Pauling. The decisive test was oxygen: MO theory placed the two highest electrons in a degenerate antibonding $\pi^*$ pair with parallel spins, predicting a paramagnetic triplet, while the tidy Lewis double bond predicted a diamagnetic molecule — and experiment sided with the orbitals. His decades systematising molecular electronic structure earned the 1966 Nobel Prize in Chemistry "for fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method." The bonding/antibonding split, the g/u and $\sigma/\pi$ labels, and the aufbau reading of a diagram used throughout this article are his handwriting on the subject.

History

1927

— Heitler and London give the first quantum valence-bond treatment of the H₂ bond

1928

— Hund and Mulliken introduce molecular orbitals and the σ, π classification of molecular states

1929

— Lennard-Jones formulates the LCAO method and explains the paramagnetism of O₂

1931

— Hückel applies molecular-orbital theory to π-electron and aromatic systems

1966

— Mulliken receives the Nobel Prize in Chemistry for the molecular-orbital method

Going further

Texts. Foot, Atomic Physics (2005), Ch. 8 (molecular structure and $\mathrm{H_2^+}$ ) · Atkins & Friedman, Molecular Quantum Mechanics, 5th ed., Ch. 8 (LCAO method) · Demtröder, Molecular Physics — LCAO-MO treatment of diatomics · Herzberg, Spectra of Diatomic Molecules (term-symbol reference) · Saleh & Teich, Fundamentals of Photonics — molecular energy levels. The energies $E_\pm=(\alpha\pm\beta)/(1\pm S)$ follow from the $2\times2$ secular determinant. The configuration $(\sigma_{2s})^2(\sigma_{2s}^*)^2(\sigma_{2p})^2(\pi_{2p})^4(\pi_{2p}^*)^2$ gives bond order 2 and a $^3\Sigma_g^-$ triplet. Dissociation energies follow the bond-order trend: $\mathrm{N_2}\approx945$ , $\mathrm{O_2}\approx498$ , $\mathrm{F_2}\approx158$ kJ/mol. The second-row orbital ordering is $\sigma_{2p}<\pi_{2p}$ for $\mathrm{O_2}$ and $\mathrm{F_2}$ , but the $s$ – $p$ -mixed order $\pi_{2p}<\sigma_{2p}$ applies for $\mathrm{Li_2}$ – $\mathrm{N_2}$ .

Prerequisites. The Born–Oppenheimer Approximation · The Chemical Bond and Molecular Potential Energy Curves. Leads to. Molecular Term Symbols · Electronic Transitions in Molecules · The Franck–Condon Principle · Selection Rules for Molecular Transitions.

Ask the tutor: Ask me to walk through the MO diagram of any second-row diatomic — say $\mathrm{B_2}$ , $\mathrm{C_2}$ , or $\mathrm{CO}$ — and predict its bond order, magnetism, and ground-state term from scratch.

Is a molecular orbital a real thing or just a mathematical convenience?

It is a genuine one-electron state of the whole molecule, with one caveat: the exact many-electron wavefunction is not literally a product of orbitals, because electrons repel and correlate. The molecular orbital is the best independent-particle description — each electron moving in the average field of the nuclei and the other electrons. What makes it more than bookkeeping is that its symmetry labels, nodal structure, and energy ordering are measurable: photoelectron spectroscopy reads orbital binding energies directly, and $\mathrm{O_2}$ 's predicted paramagnetism is a hard fact. An MO is approximate the way "the electron's energy level" is — an excellent, testable model, not a fiction.

Why does the antibonding orbital rise more than the bonding orbital falls?

Because the overlap $S$ is positive and sits in the denominators of $E_\pm=(\alpha\pm\beta)/(1\pm S)$ . The bonding shift carries a $(1+S)$ denominator and the antibonding a $(1-S)$ ; since $1-S<1+S$ , the same numerator gives a larger antibonding rise than bonding fall — the difference is proportional to $2S/(1-S^2)>0$ . Physically, forcing a node between the nuclei costs kinetic energy and leaves the nuclei poorly screened, which is why filling both members of a pair is net destabilising and why $\mathrm{He_2}$ does not bind.

Do the g/u labels apply to every molecule?

No — only to centrosymmetric ones: homonuclear diatomics and other molecules with an inversion centre. The g/u (gerade/ungerade) label is the parity of the orbital under inversion through the molecular centre, an operation that is a symmetry only when the two ends are identical. For a heteronuclear molecule like CO or HF there is no inversion centre, so the labels do not exist and orbitals are classified by the remaining symmetries. This matters for spectroscopy: the Laporte parity rule $g\leftrightarrow u$ that forbids certain transitions in $\mathrm{O_2}$ or $\mathrm{N_2}$ has no analogue there.

How can pulling an electron out of O₂ make its bond stronger?

It follows from where the electron lives. The highest occupied orbital of $\mathrm{O_2}$ is an antibonding $\pi_{2p}^*$ , and an electron there works against the bond — it adds density to the node region and raises the energy. Removing it lowers the antibonding count, so the bond order rises from 2 to 2.5 and the bond shortens (from $1.21$ Å to $1.12$ Å) and stiffens. The rule is general: adding electrons to bonding orbitals or removing them from antibonding ones strengthens a bond; the reverse weakens it — and a simple shared-pair picture has no natural way to see this.

What is the difference between MO theory and valence-bond theory?

They are two approximations to the same molecule, from opposite ends. Valence-bond theory builds a bond from a localised shared pair, one electron from each atom, close to the chemist's lines-between-atoms intuition; MO theory builds delocalised orbitals over the whole molecule and fills them like a miniature periodic table. They often agree, but VB is transparent for localised bonds and hybridisation while MO is the natural language for delocalisation, spectroscopy, and magnetism. The classic dividing line is $\mathrm{O_2}$ , where the naive VB Lewis structure predicts a diamagnetic molecule while MO theory predicts the observed paramagnetic triplet.

Why do N₂ and O₂ have different orbital orderings?

Because of $s$ – $p$ mixing. The $\sigma_{2s}$ and $\sigma_{2p}$ orbitals share symmetry, so they mix and repel, pushing $\sigma_{2p}$ upward. For the early second-row diatomics ( $\mathrm{Li_2}$ through $\mathrm{N_2}$ ) the $2s$ and $2p$ atomic levels are close, the mixing is strong, and $\sigma_{2p}$ is pushed above the $\pi_{2p}$ pair, giving $\pi_{2p}<\sigma_{2p}$ . By oxygen the $2s$ – $2p$ gap has widened, the mixing weakens, and the natural order $\sigma_{2p}<\pi_{2p}$ is restored. Crucially this changes nothing about $\mathrm{O_2}$ 's bond order or paramagnetism — those orbitals are filled and the last electrons land in $\pi_{2p}^*$ either way.